Static light scattering

Static light scattering is a technique in physical chemistry that measures the intensity of the scattered light to obtain the average molecular weight Mw of a macromolecule like a polymer or a protein in solution. Measurement of the scattering intensity at many angles allows calculation of the root mean square radius, also called the radius of gyration Rg. By measuring the scattering intensity for many samples of various concentrations, the second virial coefficient, A2, can be calculated. Static light scattering is also commonly utilized to determine the size of particle suspensions in the sub-μm and supra-μm ranges, via the Lorenz-Mie (see Mie scattering) and Fraunhofer diffraction formalisms, respectively. For static light scattering experiments, a high-intensity monochromatic light, usually a laser, is launched into a solution containing the macromolecules. One or many detectors are used to measure the scattering intensity at one or many angles. The angular dependence is required to obtain accurate measurements of both molar mass and size for all macromolecules of radius above 1–2% of the incident wavelength. Hence simultaneous measurements at several angles relative to the direction of the incident light, known as multi-angle light scattering (MALS) or multi-angle laser light scattering (MALLS), are generally regarded as the standard implementation of static light scattering. Additional details on the history and theory of MALS may be found in multi-angle light scattering. To measure the average molecular weight directly without calibration from the light scattering intensity, the laser intensity, the quantum efficiency of the detector, and the full scattering volume and solid angle of the detector need to be known. Since this is impractical, all commercial instruments are calibrated using a strong, known scatterer like toluene since the Rayleigh ratio of toluene and a few other solvents were measured using an absolute light scattering instrument.

Theory For a light scattering instrument composed of many detectors placed at various angles, all the detectors need to respond the same way. Usually, detectors will have slightly different quantum efficiency, different gains, and are looking at different geometrical scattering volumes. In this case, a normalization of the detectors is absolutely needed. To normalize the detectors, a measurement of a pure solvent is made first. Then an isotropic scatterer is added to the solvent. Since isotropic scatterers scatter the same intensity at any angle, the detector efficiency and gain can be normalized with this procedure. It is convenient to normalize all the detectors to the 90° angle detector.

N ( θ ) = I R ( θ ) − I S ( θ ) I R ( 90 ) − I S ( 90 ) {\displaystyle \ N(\theta )={\frac {I_{R}(\theta )-I_{S}(\theta )}{I_{R}(90)-I_{S}(90)}}}

where IR(90) is the scattering intensity measured for the Rayleigh scatterer by the 90° angle detector. The most common equation to measure the weight-average molecular weight, Mw, is the Zimm equation (the right-hand side of the Zimm equation is provided incorrectly in some texts, as noted by Hiemenz and Lodge):

K c Δ R ( θ , c ) = 1 M w ( 1 + q 2 R g 2 3 + O ( q 4 ) ) + 2 A 2 c + O ( c 2 ) {\displaystyle {\frac {Kc}{\Delta R(\theta ,c)}}={\frac {1}{M_{w}}}\left(1+{\frac {q^{2}R_{g}^{2}}{3}}+O(q^{4})\right)+2A_{2}c+O(c^{2})}

where

K = 4 π 2 n 0 2 ( d n / d c ) 2 / N A λ 4 {\displaystyle \ K=4\pi ^{2}n_{0}^{2}(dn/dc)^{2}/N_{\text{A}}\lambda ^{4}}

and

Δ R ( θ , c ) = R A ( θ ) − R 0 ( θ ) {\displaystyle \ \Delta R(\theta ,c)=R_{A}(\theta )-R_{0}(\theta )}

with

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